Wandering vector multipliers for unitary groups

Authors:
Deguang Han and D. Larson

Journal:
Trans. Amer. Math. Soc. **353** (2001), 3347-3370

MSC (2000):
Primary 46L10, 46L51, 42C40

DOI:
https://doi.org/10.1090/S0002-9947-01-02795-7

Published electronically:
April 9, 2001

MathSciNet review:
1828609

Full-text PDF

Abstract | References | Similar Articles | Additional Information

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system into itself. A special case of unitary system is a discrete unitary group. We prove that for many (and perhaps all) discrete unitary groups, the set of wandering vector multipliers is itself a group. We completely characterize the wandering vector multipliers for abelian and ICC unitary groups. Some characterizations of special wandering vector multipliers are obtained for other cases. In particular, there are simple characterizations for diagonal and permutation wandering vector multipliers. Similar results remain valid for irrational rotation unitary systems. We also obtain some results concerning the wandering vector multipliers for those unitary systems which are the ordered products of two unitary groups. There are applications to wavelet systems.

**[BKM]**V. Bergelson, I. Kornfeld, and B. Mityagin,*Weakly wandering vectors for unitary action of commutative groups,*J. Funct. Anal.,**126**(1994), 274-304. MR**96c:47008****[BR]**O. Bratteli and D. W. Robinson,*Operator Algebras and Quantum Statistical Mechanics I,*Springer-Verlag, 1987. MR**98e:82004****[DL]**X. Dai and D. Larson,*Wandering vectors for unitary systems and orthogonal wavelets,*Memoirs Amer. Math. Soc.,**134**(1998), no. 640. MR**98m:47067****[DGLL]**X. Dai, Q, Gu, D. Larson and R. Liang,*Wavelet multipliers and the phase of an orthonormal wavelet,*preprint.**[Di]**J. Diximer,*Von Neumann Algebras,*North-Holland, 1981. MR**83a:46004****[Ha]**D. Han,*Wandering vectors for irrational rotation unitary systems,*Trans. Amer. Math. Soc.,**350**(1998), 321-329. MR**98k:47089****[JR]**N. Jacobson and C. Rickart,*Jordan homomorphisms of rings,*Tran. Amer. Math. Soc.,**69**(1950), 479-502. MR**12:387h****[Ka]**R. Kadison,*Isometries of operator algebras,*Ann. of Math.,**54**(1951), 325-338. MR**13:256a****[KR]**R. Kadison and J. Ringrose,*``Fundamentals of the Theory of Operator Algebras,''*vols. I and II, Academic Press, Orlando, 1983 and 1986. MR**85j:46099**; MR**88d:46106****[LMT]**W. Li, J. McCarthy and D. Timotin,*A note on wavelets for unitary systems,*unpublished note.**[Pe]**K. Petersen,*Ergodic Theory,*Cambridge University Press, 1995. MR**95k:28003****[RR]**H. Radjavi and P. Rosenthal,*Invariant Subspaces,*Springer-Velag, 1973. MR**51:3924****[Wut]**The Wutam Consortium,*Basic properties of wavelets,*J. Fourier Anal. Appl.,**4**(1998), 575-594. MR**99i:42056**

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Additional Information

**Deguang Han**

Affiliation:
Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364

Email:
dhan@pegasus.cc.ucf.edu

**D. Larson**

Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843

Email:
David.Larson@math.tamu.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02795-7

Keywords:
Unitary system,
wandering vector,
wandering vector multiplier,
von Neumann algebra,
wavelet system

Received by editor(s):
February 5, 1998

Published electronically:
April 9, 2001

Additional Notes:
(DH) Participant, Workshop in Linear Analysis and Probability, Texas A&M University

(DL) This work was partially supported by NSF

Article copyright:
© Copyright 2001
American Mathematical Society